Microscopic approach to entropy production

Wittkowski, Raphael; Löwen, Hartmut; Brand, Helmut R.

doi:10.1088/1751-8113/46/35/355003

Cited by 16 publications

(27 citation statements)

References 63 publications

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“…to calculate entropy production (Wittkowski et al [2013]). For active matter (see section 6), defining and calculating the entropy is even more difficult (Nardini et al [2017]).…”

Section: First Problem: the Source Of Irreversibility In Thermodynamicsmentioning

confidence: 99%

The Five Problems of Irreversibility

Vrugt

2020

Preprint

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Thermodynamics has a clear arrow of time, characterized by the irreversible approach to equilibrium. This stands in contrast to the laws of microscopic theories, which are invariant under time-reversal. In this article, I show that the difficulty in solving this "problem of irreversibility" partly arises from the fact that it actually consists of five different subproblems. These concern the source of irreversibility in thermodynamics, the definition of equilibrium and entropy, the justification of coarse-graining, the approach to equilibrium and the arrow of time. Not distinguishing them can lead and has led to terminological confusion, apparent contradictions between positions that are actually compatible, and difficulties in connecting the debates in physics and philosophy of physics.

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“…to calculate entropy production (Wittkowski et al [2013]). For active matter (see section 6), defining and calculating the entropy is even more difficult (Nardini et al [2017]).…”

Section: First Problem: the Source Of Irreversibility In Thermodynamicsmentioning

confidence: 99%

The Five Problems of Irreversibility

Vrugt

2020

Preprint

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“…Using the cycle observables equation (11) and inserting the cycle decomposition equation (17), the average of a current-like observable…”

Section: It Has Two Important Propertiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…But even then the resulting equations are highly non-trivial both from a physical and mathematical point of view. Within this framework attempts have been made to derive dynamical density functional theory [10] and its extension to nonequilibrium situations [11,12].Indeed, another layer of complexity is added when going away from thermal equilibrium to driven systems. Stochastic thermodynamics has emerged as a comprehensive theoretical framework in particular for driven systems that are still in contact with a heat reservoir that itself remains in equilibrium [13].…”

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confidence: 99%

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Cycle representatives for the coarse-graining of systems driven into a non-equilibrium steady state

Knoch

Speck

2015

New J. Phys.

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A major current challenge poses the systematic construction of coarse-grained models that are dynamically consistent, and, moreover, might be used for systems driven out of thermal equilibrium. Here we present a novel prescription that extends the Markov state modelling approach to driven systems. The first step is to construct a complex network of microstates from detailed atomistic simulations with transition rates that break detailed balance. The coarse-graining is then carried out in the cycle space of this network. To this end we introduce the concept of representatives, which stand for many cycles with similar properties. We show how to find these cycle communities using welldeveloped standard algorithms. Removing all cycles except for the representatives defines the coarse-grained model, which is mapped back onto a network with far fewer states and renormalized transition rates that, however, preserve the entropy production of the original network. Our approach is illustrated and validated for a single driven particle.

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“…This allows to derive mesoscopic and macroscopic theories based on known microscopic equations of motion in a systematic and rather compact way of coarse graining [11][12][13]. Therefore, the Mori-Zwanzig formalism is a very useful method for a variety of fields, such as fluid mechanics [3,5], polymer physics [11,14], classical dynamical density functional theory [15][16][17][18][19], solid-state theory [20], spin relaxation theory [9,21], dielectric relaxation theory [22,23], spectroscopy [24], calculation of correlation functions [6], plasma physics [25], and particle physics [26]. Due to its great importance, it is also studied in disciplines outside of physics such as mathematics [10,27,28] and philosophy [29].…”

Section: Introductionmentioning

confidence: 99%

Mori-Zwanzig projection operator formalism for far-from-equilibrium systems with time-dependent Hamiltonians

Vrugt

Wittkowski

2019

Phys. Rev. E

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The Mori-Zwanzig projection operator formalism is a powerful method for the derivation of mesoscopic and macroscopic theories based on known microscopic equations of motion. It has applications in a large number of areas including fluid mechanics, solid-state theory, spin relaxation theory, and particle physics. In its present form, however, the formalism cannot be directly applied to systems with time-dependent Hamiltonians. Such systems are relevant in a lot of scenarios like, for example, driven soft matter or nuclear magnetic resonance. In this article, we derive a generalization of the present Mori-Zwanzig formalism that is able to treat also time-dependent Hamiltonians. The extended formalism can be applied to classical and quantum systems, close to and far from thermodynamic equilibrium, and even in the case of explicitly time-dependent observables. Moreover, we develop a variety of approximation techniques that enhance the practical applicability of our formalism. Generalizations and approximations are developed for both equations of motion and correlation functions. Our formalism is demonstrated for the important case of spin relaxation in a time-dependent external magnetic field. The Bloch equations are derived together with microscopic expressions for the relaxation times.The currently used form of the Mori-Zwanzig formalism faces the problem that it cannot be directly applied to systems with time-dependent Hamiltonians. Those, however, are relevant for a lot of scenarios including soft matter systems subject to time-dependent external driving forces [30,31] or nuclear magnetic resonance (NMR) measurements with rapidly varying electromagnetic pulses [21].If arising, time-dependent Hamiltonians can sometimes be treated as additional external perturbations [3]. This requires, however, that the perturbation is sufficiently small and couples to the macroscopic variables only. Generalizations of the projection operator method towards non-Hamiltonian dynamical systems have been developed by Chorin, Hald, and Kupferman [27,32]. Xing and Kim use mappings between dissipative and Hamiltonian systems [33] to apply projection operators in the non-Hamiltonian case [34]. The methods from Refs. [32][33][34], however, are not applicable to quantum-mechanical systems. Moreover, approximation methods commonly applied in the context of the Mori-Zwanzig formalism, such as the linearization around thermodynamic equilibrium [3], rely on the existence of a Hamiltonian.The Mori-Zwanzig formalism exists in a variety of forms. The original theory developed by Mori [1] can be applied to systems with time-independent Hamiltonians that are close to thermal equilibrium. A generalization towards systems far from equilibrium using timedependent projection operators has been presented by Robertson [35], Kawasaki and Gunton [36], and Grabert [3,7]. Recently, Bouchard has derived an extension of the Mori theory towards systems with time-dependent Hamiltonians [21] that can be applied close to equilibrium. What is still missing, however...

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Microscopic approach to entropy production

Cited by 16 publications

References 63 publications

The Five Problems of Irreversibility

The Five Problems of Irreversibility

Cycle representatives for the coarse-graining of systems driven into a non-equilibrium steady state

Mori-Zwanzig projection operator formalism for far-from-equilibrium systems with time-dependent Hamiltonians

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